How are functions used to describe relationships that happen in our lives?
Essential Questions:
How do you represent a function with a table, a graph, and a rule?
Standards
8. F.1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (Function notation is not required in Grade 8.)
Objectives
The student will use the concept of an input/out machine to demonstrate their understanding of mathematical relationship/function by generating rate tables.
The student will generate a rule (verbally and/or algebraically) to describe a mathematical relationship.
The student will use their rate table to produce a visual representation of their data by creating a graph.
The student will identify if a relationship is a function or non-function given a table or graph.
Reflection and/or Comments from your PCSSD 8th Grade Curriculum Team
We hopefully have provided enough time to make the six out of eight Mathematical Practices part of your children’s approach to mathematics. We have waited to allow time for the Practices to blend and intertwine. This unit is an excellent representation of Mathematical Practices #4. Students will be required to look at relations and functions from concrete models or pictures; from verbal description; from graphical displays; and from symbolic algebraic notation. The students will need to move from one representation to another. They will need to compare different situations from different perspectives: graphical, verbal, and/or algebraic.
Mathematical Practice #1 will be a perfect companion to Mathematical Practice #4. Students will analyze problems with various representations, and develop a plan to obtain the solution.
In grade 6, students plotted points in all four quadrants of the coordinate plane. They also represented and analyzed quantitative relationships between dependent and independent variables. In Grade 7, students decided whether two quantities are in a proportional relationship.
In Grade 8, students begin to call relationships functions when each input is assigned to exactly one output. Also, in Grade 8, students learn that proportional relationships are part of a broader group of linear functions, and they are able to identify whether a relationship is linear. Nonlinear functions are included for comparison. Later, in high school, students use function notation and are able to identify types of nonlinear functions.
To determine whether a relationship is a function, students should be expected to reason from a context, a graph, or a table, after first being clear which quantity is considered the input and which is the output. When a relationship is not a function, students should produce a counterexample: an “input value” with at least two “output values.” If the relationship is a function, the students should explain how they verified that for each input there was exactly one output. The “vertical line test” should be avoided because
it is too easy to apply without thinking,
students do not need an efficient strategy at this point, and
It creates misconceptions for later mathematics, when it is useful to think of functions more broadly, such as whether x might be a function of y.
“Function machine” pictures are useful for helping students imagine input and output values, with a rule inside the machine by which the output value is determined from the input.
Notice that the standards explicitly call for exploring functions numerically, graphically, verbally, and algebraically (symbolically, with letters). This is sometimes called the “rule of four.” For fluency and flexibility in thinking, students need experiences translating among these.
In Grade 8, the focus, of course, is on linear functions, and students begin to recognize a linear function from its form y = mx + b. Students also need experiences with nonlinear functions, including functions given by graphs, tables, or verbal descriptions but for which there is no formula for the rule, such as a girl’s height as a function of her age.
Assessment Product
Students will create a portfolio detailing their growth pertaining to the topics addressed throughout this unit on functions. The Entries provided follow the theme of construction repair and remodel. At the end of the unit, students will be grouped and assigned an Entry. Each group member will use the work in their portfolio for that assigned Entry to prepare a class presentation. They will be encouraged to enhance the depth of the problem assigned. (Every week in the unit, a new Entry will be provided.)
Entry #1
The local carpenter went to Home Depot to buy nails. He needed nails for three separate projects: 2lbs for $1.20, 3lbs for $1.80, and 4lbs for $2.40.
Identify the input value, independent variable, or the domain of the function. Explain the vocabulary.
Identify the output value, dependent variable, or the range of the function. Explain the vocabulary.
Create a table.
Is this relation a function? Why?
Use the pattern in the table to write a rule. Identify the variables being used.
If the carpenter buys 10lbs, how much will he pay? Justify your answer by using the table and applying the rule. Show and explain.
Graph the results from the table on a coordinate plane.
Key Questions
What do the numbers that go into the machine represent? Input, Independent variable, Domain, x-values
What do the numbers that come out of the machine represent? Output, dependent variable, Range, y-values
Where do you get the information to produce the table? Information provided in the problem
What is the similarity and difference between the definition of a relation and a function?
Given a table, how do you know the relation is a function or not a function? For every value in the domain there is only one value in the range (the x-value does not repeat)
Given a graph, how do you know the relation is a function or not a function? It produces a linear function.
Observable Student Behaviors
Given a table or graph, students can identify and/or explain why it is a function or a non-function.
Students will start analyzing using the rule of four: numerical, graphical, verbal and algebraic interpretation of functions.
Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.
Vocabulary Function, input, output, domain, range, independent variable, dependent variables, coordinate plane, vertical and horizontal axis, ordered pairs, linear function, linear equation, constant rate of change
Suggested Activities [see Legend to highlight MCO and HYS]
Houghton Mifflin OnCore Mathematics Middle School Grade 8 Unit 2-1 (TG p.35, Student p.35-38)
ABC Mastering the Common Core in Mathematics Chapter 7.4 & 7.5 (p. 94-95)
Gizmo Lessons
Function Machines 1 (Functions and Tables)
Drop a number into a function machine, and see what number comes out! You can use one of the six pre-set function machines, or program your own function rule into one of the blank machines. Stack up to three function machines together. Input and output can be recorded in a table and on a graph. (Student Exploration Sheet & Teacher Guide Available)
Function Machines 2 (Functions, Tables, and Graphs)
Drop a number into a function machine, and see what number comes out! You can use one of the six pre-set function machines, or program your own function rule into one of the blank machines. Stack up to three function machines together. Input and output can be recorded in a table and on a graph. (Student Exploration Sheet & Teacher Guide Available)
Introduction to Functions
Determine if a relation is a function using the mapping diagram, ordered pairs, or the graph of the relation. Drag arrows from the domain to the range, type in ordered pairs, or drag points to the graph to add inputs and outputs to the relation.
Points, Lines, and Equations
Compare the graph of a linear function to its rule and to a table of its values. Change the function by dragging two points on the line. Examine how the rule and table change. (Student Exploration Sheet & Teacher Guide Available)
Teaching the Common Core Math Standards with Hands-On Activities by Judith Muschla
8.F.1 Activity 1- p. 183, Activity 2- p. 184
JBHM Grade 8 GP2 (p.265-296)
Glencoe Pre Algebra 8-1 (p.369-373) Functions
Glencoe Pre Algebra 8-4 & 8-5 (p. 387-397) Slope and Constant Rate of Change
Glencoe Algebra 1 4-6 (p.226-231) Functions
Glencoe Algebra 1 5-1 (p. 256-262) Slope
The Illustrative Mathematics Project offers guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students will experience in a faithful implementation of the Common Core State Standards. The website features a clickable version of the Common Core in mathematics and the first round of "illustrations" of specific standards with associated classroom tasks and solutions.
Diverse Learners
Odyssey (teacher discretion)
Skills Tutor (teacher discretion)
Algebra’scool: Unit C Module 9
Grade: 8 Unit: 3 Week: 1 Dates: 11/26-11/30
Content: Functions, Tables, and Graphs
Theme Essential Question:
Essential Questions:
Standards
Objectives
Reflection and/or Comments from your PCSSD 8th Grade Curriculum Team
We hopefully have provided enough time to make the six out of eight Mathematical Practices part of your children’s approach to mathematics. We have waited to allow time for the Practices to blend and intertwine.
This unit is an excellent representation of Mathematical Practices #4. Students will be required to look at relations and functions from concrete models or pictures; from verbal description; from graphical displays; and from symbolic algebraic notation. The students will need to move from one representation to another. They will need to compare different situations from different perspectives: graphical, verbal, and/or algebraic.
Mathematical Practice #1 will be a perfect companion to Mathematical Practice #4. Students will analyze problems with various representations, and develop a plan to obtain the solution.
Background Information
Recommended: For a quick overview of the standard(s) to be addressed in this lesson, see Arizona’s Content Standards Reference Materials.
**http://www.azed.gov/wp-content/uploads/PDF/MathGr8.pdf**
(Taken from **Ohio Dept of Education Mathematics Model Curriculum 6-28-2011** )
In grade 6, students plotted points in all four quadrants of the coordinate plane. They also represented and analyzed quantitative relationships between dependent and independent variables. In Grade 7, students decided whether two quantities are in a proportional relationship.
In Grade 8, students begin to call relationships functions when each input is assigned to exactly one output. Also, in Grade 8, students learn that proportional relationships are part of a broader group of linear functions, and they are able to identify whether a relationship is linear. Nonlinear functions are included for comparison. Later, in high school, students use function notation and are able to identify types of nonlinear functions.
To determine whether a relationship is a function, students should be expected to reason from a context, a graph, or a table, after first being clear which quantity is considered the input and which is the output. When a relationship is not a function, students should produce a counterexample: an “input value” with at least two “output values.” If the relationship is a function, the students should explain how they verified that for each input there was exactly one output. The “vertical line test” should be avoided because
it is too easy to apply without thinking,
students do not need an efficient strategy at this point, and
It creates misconceptions for later mathematics, when it is useful to think of functions more broadly, such as whether x might be a function of y.
“Function machine” pictures are useful for helping students imagine input and output values, with a rule inside the machine by which the output value is determined from the input.
Notice that the standards explicitly call for exploring functions numerically, graphically, verbally, and algebraically (symbolically, with letters). This is sometimes called the “rule of four.” For fluency and flexibility in thinking, students need experiences translating among these.
In Grade 8, the focus, of course, is on linear functions, and students begin to recognize a linear function from its form y = mx + b. Students also need experiences with nonlinear functions, including functions given by graphs, tables, or verbal descriptions but for which there is no formula for the rule, such as a girl’s height as a function of her age.
Assessment
Product
- Students will create a portfolio detailing their growth pertaining to the topics addressed throughout this unit on functions. The Entries provided follow the theme of construction repair and remodel. At the end of the unit, students will be grouped and assigned an Entry. Each group member will use the work in their portfolio for that assigned Entry to prepare a class presentation. They will be encouraged to enhance the depth of the problem assigned. (Every week in the unit, a new Entry will be provided.)
Entry #1Key Questions
Observable Student Behaviors
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Vocabulary
Function, input, output, domain, range, independent variable, dependent variables, coordinate plane, vertical and horizontal axis, ordered pairs, linear function, linear equation, constant rate of change
Suggested Activities [see Legend to highlight MCO and HYS]
- Houghton Mifflin OnCore Mathematics Middle School Grade 8 Unit 2-1 (TG p.35, Student p.35-38)
- ABC Mastering the Common Core in Mathematics Chapter 7.4 & 7.5 (p. 94-95)
- Gizmo Lessons
- Function Machines 1 (Functions and Tables)
Drop a number into a function machine, and see what number comes out! You can use one of the six pre-set function machines, or program your own function rule into one of the blank machines. Stack up to three function machines together. Input and output can be recorded in a table and on a graph. (Student Exploration Sheet & Teacher Guide Available)- Function Machines 2 (Functions, Tables, and Graphs)
Drop a number into a function machine, and see what number comes out! You can use one of the six pre-set function machines, or program your own function rule into one of the blank machines. Stack up to three function machines together. Input and output can be recorded in a table and on a graph. (Student Exploration Sheet & Teacher Guide Available)- Introduction to Functions
Determine if a relation is a function using the mapping diagram, ordered pairs, or the graph of the relation. Drag arrows from the domain to the range, type in ordered pairs, or drag points to the graph to add inputs and outputs to the relation.- Points, Lines, and Equations
Compare the graph of a linear function to its rule and to a table of its values. Change the function by dragging two points on the line. Examine how the rule and table change. (Student Exploration Sheet & Teacher Guide Available)- Teaching the Common Core Math Standards with Hands-On Activities by Judith Muschla
- 8.F.1 Activity 1- p. 183, Activity 2- p. 184
- JBHM Grade 8 GP2 (p.265-296)
- Glencoe Pre Algebra 8-1 (p.369-373) Functions
- Glencoe Pre Algebra 8-4 & 8-5 (p. 387-397) Slope and Constant Rate of Change
- Glencoe Algebra 1 4-6 (p.226-231) Functions
- Glencoe Algebra 1 5-1 (p. 256-262) Slope
The Illustrative Mathematics Project offers guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students will experience in a faithful implementation of the Common Core State Standards. The website features a clickable version of the Common Core in mathematics and the first round of "illustrations" of specific standards with associated classroom tasks and solutions.Diverse Learners
Odyssey (teacher discretion)
Skills Tutor (teacher discretion)
Algebra’scool: Unit C Module 9
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